JEE Mathematics 2025 Question with Solution

Given: The curve $y = f(x)$ passes through $(0,5)$ and $(\log 2, k)$ and satisfies

$$2(3+y)e^{2x}dx - (7+e^{2x})dy = 0$$

Find: The value of $k$.

Rearrange the differential equation and separate variables:

$$2(3+y)e^{2x} \, dx = (7+e^{2x}) \, dy$$ $$\frac{dy}{2(3+y)} = \frac{e^{2x} \, dx}{7+e^{2x}}$$

Integrate both sides:

$$\int \frac{dy}{2(3+y)} = \int \frac{e^{2x}}{7+e^{2x}} \, dx$$

Hence,

$$\frac{1}{2} \ln|3+y| = \int \frac{e^{2x}}{7+e^{2x}} \, dx$$

For the right side, use the substitution $t = 7 + e^{2x}$. Then

$$dt = 2e^{2x} \, dx$$

so

$$e^{2x} \, dx = \frac{1}{2}dt$$

Therefore,

$$\int \frac{e^{2x}}{7+e^{2x}} \, dx = \frac{1}{2} \int \frac{1}{t} \, dt = \frac{1}{2}\ln|t| = \frac{1}{2}\ln|7+e^{2x}|$$

So the integrated equation is

$$\frac{1}{2} \ln|3+y| = \frac{1}{2} \ln|7+e^{2x}| + C$$

Multiplying by $2$,

$$\ln|3+y| = \ln|7+e^{2x}| + C$$

Exponentiating,

$$|3+y| = A|7+e^{2x}|$$

where $A = e^C$.

Use the point $(0,5)$:

$$|3+5| = A|7+e^0|$$ $$8 = A(8)$$

So,

$$A = 1$$

Thus,

$$3+y = 7+e^{2x}$$

which gives

$$y = 4 + e^{2x}$$

Now substitute $x = \log 2$:

$$k = y = 4 + e^{2\log 2} = 4 + 2^2 = 8$$

Therefore, the value of $k$ is $8$ and the correct option is B.

Given:

$$2(3+y)e^{2x}dx - (7+e^{2x})dy = 0$$

Find: The value of $k$.

Write the equation in derivative form:

$$\frac{dy}{dx} = \frac{2(3+y)e^{2x}}{7+e^{2x}}$$

This can be rewritten as

$$\frac{dy}{dx} - \frac{2e^{2x}}{7+e^{2x}}y = \frac{6e^{2x}}{7+e^{2x}}$$

Treating it as a linear differential equation, the integrating factor used in the provided solution is

$$I.F. = e^{-\int \frac{2e^{2x}dx}{7+e^{2x}}} = \frac{1}{7+e^{2x}}$$

Multiplying through by the integrating factor:

$$\frac{y}{7+e^{2x}} = \int \frac{6e^{2x}dx}{(7+e^{2x})^2}$$

Integrating gives

$$\frac{y}{7+e^{2x}} = \frac{-3}{7+e^{2x}} + C$$

Apply the condition $(0,5)$:

$$\frac{5}{8} = \frac{-3}{8} + C$$

So,

$$C = 1$$

Hence,

$$y = -3 + 7 + e^{2x} = e^{2x} + 4$$

At $x = \log 2$,

$$k = e^{2\log 2} + 4 = 4 + 4 = 8$$

Therefore, the correct option is B.